Rigid Body Simulator Calculator

Inputs

Inertia source

Choose geometry or enter principal moments.

Shape

Assumes principal axes through the center.

Mass m (kg)

Used only for shape-based inertia.

Length L (m)

Width W (m)

Height H (m)

Ix (kg·m²)

Iy (kg·m²)

Iz (kg·m²)

Torque τx (N·m)

Torque τy (N·m)

Torque τz (N·m)

Initial ωx (rad/s)

Initial ωy (rad/s)

Initial ωz (rad/s)

Initial roll (deg)

Initial pitch (deg)

Initial yaw (deg)

Time step dt (s)

Smaller dt improves stability and accuracy.

Duration T (s)

Output every (s)

Table spacing; not the integration step.

Integrator

RK4 is smoother for many cases.

Scenario	Body	m (kg)	Dimensions (m)	τ (N·m)	ω0 (rad/s)	dt, T (s)	Expected behavior
Free precession	Block	2.0	L=0.40, W=0.20, H=0.10	(0, 0, 0)	(0, 3, 0)	0.01, 3	ω shifts due to asymmetric inertia.
Spin-up	Cylinder	1.5	r=0.08, h=0.20	(0, 0, 0.15)	(0, 0, 2)	0.005, 2	Yaw accelerates with τz applied.
Stable sphere	Sphere	1.0	r=0.10	(0.05, 0, 0)	(0, 0, 0)	0.01, 2	All axes behave similarly for equal inertia.

This simulator assumes the rigid body rotates about its principal axes. The inertia tensor is diagonal, with principal moments Ix, Iy, and Iz.

Euler’s rotational equations for constant applied torque τ are:

Orientation is updated using a unit quaternion q with:

q̇ = 0.5 · q ⊗ [0, ω]

Angular momentum and energy are reported as: L = Iω and E = 0.5 (Ix ωx² + Iy ωy² + Iz ωz²).